### Mathematical analysis of atomistic fracture and near-crack-tip plasticity in crystalline materials

Fluids and Materials Seminar

3rd February 2022, 2:00 pm – 3:00 pm

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The modelling of atomistic fracture and related phenomena in crystalline materials poses a string of mathematically non-trivial (and exciting) theoretical and practical challenges. At the heart of the problem lies a discrete domain of atoms (a lattice), which exhibits spatial inhomogeneity induced by the crack surface, particularly pronounced in the vicinity of the crack tip. Atoms interact in a highly nonlinear way, resulting in a severely non-convex energy landscape facilitating non-trivial behaviour of atoms. Two particularly interesting instances of this are (i) near-crack-tip plasticity - emission and movement of defects known as dislocations in the vicinity of the crack tip; (ii) surface effects - atoms at the crack surface relaxing or possibly attaining an altogether different crystalline structure. On the practical side, the richness of possible phenomena renders the task of setting up numerical simulations particularly tricky - numerical artefacts, e.g. induced by prescribing a particular boundary condition, can fundamentally alter the results.

In this talk I will summarise on-going efforts aimed at putting the atomistic modelling of fracture and near-crack-tip plasticity on a rigorous mathematical footing. I will discuss a framework giving rise to well-defined models for which regularity and stability of solutions can be discussed. I will then show how the theory can also provide a guide for simulations in the form of consistent boundary conditions for which rigorous error analysis is possible. I will also outline how this framework can be used to rigorously derived upscaled models of near-crack-tip plasticity.

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